Timeline for Is $B \cap L^2\big( (0,T) \times (0,1)\big)$ closed in $L^2\big( (0,T) \times (0,1)\big)$?
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| Jun 14, 2023 at 16:14 | comment | added | Iosif Pinelis | @elmas : No, the Riesz–Fischer theorem is not used here. That the convergence in measure implies the almost everywhere convergence for a subsequence is a simple theorem, found I think in almost any textbook on measure theory -- see e.g. Theorem 2.30 on p. 61 of Folland, Real Analysis, Second Edition. (You can also find this here. More immediately here, you can use Corollary 2.32 on p. 62 of Folland's book, stating that the convergence in $L^1$ implies the almost everywhere convergence for a subsequence. | |
| Jun 14, 2023 at 13:13 | comment | added | elmas | Could you please clarify whether the transition from convergence in measure of $(g^{n})$ to almost everywhere convergence is established by employing the Riesz-Fischer theorem? | |
| Jun 14, 2023 at 1:35 | vote | accept | elmas | ||
| Jun 14, 2023 at 0:28 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jun 14, 2023 at 0:23 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |